Given two \(n \times n\) Hermitian matrices \(A\) and \(B\), H. Weyl [Math. Ann. 71, 441–479 (1912; JFM 43.0436.01)] asked for the possible sets of eigenvalues of \(A+B\), counting multiplicities. This problem, often called Horn’s problem, was studied extensively by A. Horn [Pac. J. Math. 12, 225–241 (1962; Zbl 0112.01501)] and by R. C. Thompson [“Spectrum of a sum of Hermitian matrices and singular values of a product of general matrices”, Lecture held at Johns Hopkins University (1988), http://www.math.sjsu.edu/~so/lecture6.pdf]. The support of the spectrum of \(C\) was investigated by A. A. Klyachko [Sel. Math., New Ser. 4, No. 3, 419–445 (1998; Zbl 0915.14010)] and by A. Knutson and T. Tao [J. Am. Math. Soc. 12, No. 4, 1055–1090 (1999; Zbl 0944.05097); Notices Am. Math. Soc. 48, No. 2, 175–186 (2001; Zbl 1047.15006)]. W. Fulton [Bull. Am. Math. Soc., New Ser. 37, No. 3, 209–249 (2000; Zbl 0994.15021)] provided a good summary of the problem and its development. The probability distribution function (PDF) of the eigenvalues of \(A+B\) has also been addressed if \(A\) and \(B\) are independently and uniformly distributed on the orbit under the adjoint action of the \(\mbox{SU}(n)\) group. Similar considerations, applied to the case of real skew-symmetric matrices under the adjoint action of \(\mbox{O}(n)\) and \(\mbox{SO}(n)\), are due to J.-B. Zuber [Ann. Inst. Henri Poincaré D, Comb. Phys. Interact. (AIHPD) 5, No. 3, 309–338 (2018; Zbl 1397.15008)]. Though the eigenvalues description of Horn’s problem for the case of real symmetric matrices is the same as the Hermitian case, the PDF under the action of the orthogonal group is much more intriguing since no explicit analytic formula is known for the relevant orbital integral and the existing numerical experiments reveal the occurrence of singularities where the PDF of the eigenvalues diverges.
In this paper, the authors compare three cases, real symmetric, Hermitian and self-dual quaternionic \(3\times 3\) matrices, to reproduce the main features of Horn’s problem and to analyze the location and nature of singularities in the symmetric case. They show that the computation of the PDF of the symmetric functions of the eigenvalues for traceless \(3\times 3\) matrices can be carried out in terms of algebraic functions and integrals. For a particular case, they reproduce the expected singular patterns. Moreover, they confirm numerically the connection between this PDF and the rescaled structure constants of zonal polynomials. Some nice numerical plots are provided.